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m Reverted edits by 2409:4063:421C:8EE8:3870:9398:3568:5C0 (talk) (HG) (3.4.10) |
→Basic remarks: equivalent conditions for unitarity |
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* [[skew-Hermitian matrix|skew Hermitian]] or antihermitian if <math>\boldsymbol{A}=-\boldsymbol{A}^\mathrm{H}</math>; i.e., <math>a_{ij} = -\overline{a_{ji}}</math> .
* [[normal matrix|normal]] if <math>\boldsymbol{A}^\mathrm{H} \boldsymbol{A} = \boldsymbol{A} \boldsymbol{A}^\mathrm{H}</math>.
* [[Unitary matrix|unitary]] if <math>\boldsymbol{A}^\mathrm{H} = \boldsymbol{A}^{-1}</math>, equivalently <math>\boldsymbol{A}\boldsymbol{A}^\mathrm{H} = \boldsymbol{I}</math>, equivalently <math>\boldsymbol{A}^\mathrm{H}\boldsymbol{A} = \boldsymbol{I}</math>.
Even if <math>\boldsymbol{A}</math> is not square, the two matrices <math>\boldsymbol{A}^\mathrm{H}\boldsymbol{A}</math> and <math>\boldsymbol{A}\boldsymbol{A}^\mathrm{H}</math> are both Hermitian and in fact [[positive-definite matrix|positive semi-definite matrices]].
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